A New Proof of Serre's Homological Characterization of Regular Local Rings

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A NEW PROOF OF SERRE’S HOMOLOGICAL CHARACTERIZATION OF REGULAR LOCAL RINGS

RAVI JAGADEESAN AND AARON LANDESMAN

Abstract. We give a new proof of Serre’s result that a Noetherian local ring is regular if and only if it has ﬁnite global dimension. Our proof avoids the explicit construction of a Koszul complex.

1. Introduction The proof that localization at a prime preserves regularity relies on Serre’s homological characterization of regularity (see [1, Théorème5]). This was the first application of homological methods to prove a non-homological result and inspired the use of homological techniques in algebra. In this paper, we present a new proof of Serre’s theorem using basic notions from the theory of derived categories. By a ring, we mean a commutative ring with a unit. Let dim A denote the Krull dimension of a ring A. Recall that a Noetherian local ring (A, m, k) with maximal 2 ideal m and residue field k is said to be regular if dimk m/m = dim A. Let gl. dim A denote the global dimension of A. An alternate proof of the following theorem is the main result of this paper. Theorem 1.1 (Serre [1, Théorème3]). Let A be a Noetherian local ring. The following are equivalent: (1) A is regular; (2) gl. dim A = dim A; (3) gl. dim A < ∞. Serre’s original proof of Theorem 1.1 shows the difficult step, namely that (3) implies (1), as follows. One first proves that gl. dim A < ∞ implies that gl. dim A ≤ dim A using the interaction between regular sequences and projective dimension. 2 One then uses the Koszul complex to prove that dimk m/m ≤ gl. dim A. The chain 2 of inequalities gl. dim A ≤ dim A ≤ dimk m/m ≤ gl. dim A then implies the result. The argument also shows that (3) implies (2). We prove Theorem 1.1 without having to construct a Koszul complex, adapting an approach suggested by Dennis Gaitsgory. We first prove the existence of an element f ∈ m \ m2 that is not a zero-divisor whenever m is not associated. By L proving k ⊗A A/f splits in D(A/f), we prove that gl. dim A = gl. dim A/f +1. This allows us to deal with the case of A regular and the case of 0 < gl. dim A < ∞ uniformly to conclude the proof of Theorem 1.1.

Notation and conventions. All complexes are graded homologically. For a ring L R, let D(R) denote the derived category of R-modules. Let − ⊗R − denote the derived tensor product. 1 2 RAVI JAGADEESAN AND AARON LANDESMAN

2. Proof of Theorem 1.1 The proof relies on the following technical result whose proof is deferred to Section3. Proposition 2.1. Let (A, m, k) be a local ring. If f ∈ m \ m2 is not a zero-divisor, then gl. dim A = gl. dim A/f + 1. In proving Serre’s theorem, we will use the following easy fact. Lemma 2.2. If A is a Noetherian local ring with gl. dim A = 0, then A is a field. Proof. Because gl. dim A = 0, every finitely generated A-module is projective. Be- cause A is local, every finitely generated projective A-module is free. Therefore, every finitely generated A-module is free, which implies that A is a field. In order to apply Proposition 2.1 in the proof of the inductive step, we will need the following two lemmas. The former is a form of prime avoidance, where one of the ideals is taken to be m2, and the rest are taken to be the associated prime ideals, whose proof can be found in [2, Tag 00DS]). The latter is immediate from [1, Proposition 4], but we provide a proof for completeness. Proposition 2.3 (Prime Avoidance [2, Tag 00DS]). Let (A, m) be a Noetherian local ring. If m is not associated, then there exists f ∈ m \ m2 that is not a zero- divisor. Lemma 2.4. If (A, m) is a Noetherian local ring with 0 < gl. dim A < ∞, then m is not an associated prime of A. Proof. Let n = gl. dim A. Assume for sake of deriving a contradiction that m is associated. There exists an element a ∈ A with Ann(a) = m, so that there exists a nonzero homomorphism θ ∈ Hom(k, A) sending 1 to a. Consider the long exact sequence of Tor associated to the short exact sequence 0 k A M 0 where M = coker θ. We obtain an exact sequence

A A A Torn+1(M, k) Torn (k, k) Torn (A, k). . The first term vanishes because gl. dim A = n and the third term vanishes because A is projective and n > 0. Therefore, the middle term vanishes as well. This A contradicts the fact that Torn (k, k) 6= 0 for a Noetherian local ring A of global dimension n with residue field k. We are now ready to prove Serre’s theorem. Proof of Theorem 1.1. It is clear that (2) implies (3). We will prove that (1) implies (2) and that (3) implies (1). Step 1: (1) =⇒ (2). We proceed by induction on dim A. Suppose first that A is a regular local ring with dim A = 0. Then, A is a field, so gl. dim A = 0. We now prove the inductive step. Let n be a positive integer and assume that gl. dim A = n − 1 for all regular local rings A with dim A = n − 1. Let (A, m) be a regular local ring with dim A = n. Because dim A ≥ 1, there exists an element A NEW PROOF OF SERRE’S HOMOLOGICAL CHARACTERIZATION OF REGULARITY 3 f ∈ m \ m2. Because A is regular local, it an integral domain, so f is not a zero- divisor. Krull’s principal ideal theorem ensures that A/f is a regular local ring of Krull dimension n − 1. The induction hypothesis guarantees that gl. dim A = n − 1. Proposition 2.1 yields that gl. dim A = n, which completes the proof of the inductive step.

Step 2: (3) =⇒ (1). We proceed by induction on gl. dim A. Suppose ﬁrst that gl. dim A = 0. Lemma 2.2 implies that A is a ﬁeld, hence regular. We now prove the inductive step. Let n be a positive integer and assume that all Noetherian local rings A with gl. dim A = n − 1 are regular. Let (A, m) be a Noetherian local ring with gl. dim A = n. Proposition 2.3 and Lemma 2.4 ensure that there exists an element f ∈ m \ m2 that is not a zero-divisor. Proposition 2.1 implies that gl. dim A/f = n − 1. The inductive hypothesis guarantees that A/f is regular. Because f ∈ m is not a zero-divisor, the local ring A is regular as well, which completes the proof of the inductive step.

3. Proof of Proposition 2.1 The key technical tool in the proof of Proposition 2.1 is the following lemma. Recall that if R is a ring, we use D(R) to denote the category of derived R modules. Lemma 3.1. Let (A, m) be a local ring. If f ∈ m \ m2 is not a zero-divisor, then L k ⊗A A/f and k ⊕ k[1] are isomorphic in D(A/f). Proof of Proposition 2.1 assuming Lemma 3.1. Recall the isomorphism L ∼ L L k ⊗A k = k ⊗A A/f ⊗A/f k in D(A/f) (see, for example, [2, Tag 06Y5]). Lemma 3.1 yields that

L ∼ L L ∼ L L L k ⊗A k = k ⊗A A/f ⊗A/f k = (k ⊕ k[1]) ⊗A/f k = k ⊗A/f k ⊕ k ⊗A/f k [1]. Taking homology yields that A ∼ A/f A Tori (k, k) = Tori−1 (k, k) ⊕ Tori (k, k) for all i. Recall the equality

B gl. dim B = sup{i | Tori (k, k) 6= 0} in a Noetherian local ring B with residue ﬁeld k. It follows that

A n A/f o gl. dim A = sup i | Tori (k, k) 6= 0 = 1 + sup i | Tori (k, k) 6= 0 = 1 + gl. dim A/f, as desired.

The strategy of the proof of Lemma 3.1 is to ﬁnd a two-term complex C• of A/f- L modules representing k ⊗A A/f in D(A/f) and then produce a quasi-isomorphism from C• to k ⊕ k[1]. The description of C•, which we give in Lemma 3.2, follows A from the fact that Tori (k, A/f) = 0 for i ≥ 2. The existence of the second quasi- isomorphism relies crucially on the assumption that f∈ / m2. 4 RAVI JAGADEESAN AND AARON LANDESMAN

Lemma 3.2. Let (A, m, k) be a Noetherian local ring and suppose that f ∈ m is not a zero-divisor. The complex

i⊗AA/f C• : 0 m ⊗A A/f A ⊗A A/f 0

L represents k ⊗A A/f in D(A/f), where i : m → A is the inclusion. Furthermore, ∼ we have H1(C•) = H0(C•) = k. Proof. The short exact sequence of modules

0 m A k 0 yields an exact triple

L L L m ⊗A A/f A ⊗A A/f k ⊗A A/f in D(A/f). We will show the ﬁrst two terms of the above sequence lie in the heart L ∼ ∼ of D(A/f). Note that A ⊗A A/f = A ⊗A A/f = A/f lies in the heart. To prove L A that m ⊗A A/f lies in the heart, it suﬃces to prove that Tori (m, A/f) = 0 for all i ≥ 0. We have a resolution of A/f given by

×f F• : 0 A A A/f 0.

A Therefore, Tori (m, A/f) is the ith homology of the complex

×f 0 m m 0.

A Because f is not a zero-divisor on m, we obtain that Tor1 (m, A/f) = 0, and it A L ∼ is manifestly true that Tori (m, A/f) = 0 for i > 1. It follows that m ⊗A A/f = m ⊗A A/f lies in the heart. The discussion of the previous paragraph yields that ∼ L Cone (i ⊗A A/f) = k ⊗A A/f L in D(A/f). The complex C• represents k ⊗A A/f in D(A/f) because m ⊗A A/f and A ⊗A A/f (the domain and codomain of i ⊗A A/f) are complexes concentrated in degree 0. ∼ Finally, to see that H1(C•) = H0(C•) = k, simply tensor the resolution F• of A A/f by k and recall that Hi(C•) = Tori (k, A/f). Proof of Lemma 3.1. We ﬁrst prove the existence of a splitting θ of the canonical inclusion

ψ : H1(C•) = ker (i ⊗A A/f) m ⊗A A/f. ∼ Note that f ⊗1 ∈ ker (i ⊗A A/f) and f ⊗1 6= 0. Lemma 3.2 ensures that H1(C•) = k, so that f ⊗ 1 generates H1(C•). Let ζ be the composite ∼ 2 ζ : m ⊗A A/f = m/fm → m/m . Note that ζ(f ⊗1) 6= 0 because f∈ / m2. Because f generates the simple A/f-module 2 H1(C•), the composite ζ ◦ ψ is an injection from H1(C•) to m/m . Because both 2 2 H1(C•) and m/m are vector spaces over k, there exists a map ξ : m/m → H1(C•) splitting the inclusion ζ ◦ ψ. We can then let θ = ξ ◦ ζ to split the inclusion ψ. A NEW PROOF OF SERRE’S HOMOLOGICAL CHARACTERIZATION OF REGULARITY 5

The map θ ﬁts into a commutative square θ m ⊗A A/f H1(C•)

i⊗AA/f 0

A ⊗A A/f H0(C•) with surjective horizontal maps, which yields a quasi-isomorphism from C• to L ∼ H0(C•) ⊕ H1(C•)[1]. It follows from Lemma 3.2 that k ⊗A A/f = k ⊕ k[1] in D(A/f), as desired. Acknowledgements We would like to thank Dennis Gaitsgory for suggesting this project and for much help understanding the subtleties of this argument. We would also like to thank Justin Campbell for helpful conversations about this proof and derived categories in general.

References [1] J.-P. Serre. Sur la dimension homologique des anneaux et des modules Noeth´eriens.In Proceed- ings of the International Symposium on Algebraic Number Theory, Tokyo & Nikko, volume 1956, pages 175–189, 1955. [2] Stacks Project Authors. Stacks Project. http://stacks.math.columbia.edu, 2015.

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